Financial Engineering 6

Lecture 21

Option Pricing with Black-Scholes Part 2

References: Villalobos, Luenberger

Lecture Topics • Quick Review • Black-Scholes Put Option Pricing • Example • More on the Greeks

Assumptions for Black-Scholes • The share does not pay any dividends.

– If dividends were paid during the time that we want to price the option, then we would need to discount the dividends.

• Transactions cost and taxes are zero.

• Interest rates are constant.

• There are no penalties for short sales of stock.

• The market operates continuously, and the share prices follow a continuous Ito Process.

• The distribution of terminal stock prices (returns) is log-normal.

Value of a Call Option • From the previous lecture:

• Which is the Black-Scholes model for a European call option valuation with a strike price of K.

• According to Luenberger:

[ ] ( ) ( )

( ) [ ]

1 2 0

2

0 1 2

N N

N

N

rt rt

rt

d C d e S e K

d

S N d Ke d

−

−

  = −  

  = −

( ) ( ) ( ) ( )1 2N N r T tC S,t S d Ke d− −= −

( )

tT

tTr K S

d −

− 

  

 ++

 

  

= σ

σ 2

ln 2

1

( ) ( )tTd

tT

tTr K S

d −−= −

− 

  

 −+



  



= σ σ

σ

1

2

2 2

ln

Black-Scholes Put Option Pricing • The Call Option pricing equation is:

• The Put Option pricing equation is very similar:

( ) ( ) ( )

( ) ( ) ( )

( ) 1 2

( ) 2 1

, N N or

, N N

r T t

r T t

P S t S d Ke d

P S t Ke d S d

− −

− −

= − − + −

= − − −

( ) ( ) ( ) ( )1 2N N r T tC S,t S d Ke d− −= −

( ) 2

1 2 1

ln 2

S r T t

K d d d T t

T t

σ

σ σ

  + + −       = = − −

−

KO Example • The closing price for KO on April 3, 2007 was of $48.99. • Consider the following option for Coca Cola:

– KODJ, strike price = $50, expiration date April; third Friday is April 20)

– Trading days remaining = 13; 17 days total. – Annual interest rate of 4%. – Information provided by Yahoo:

Position Num OptSym Expire Days Strike Type IV Vol OI Buy 1 KODJ 7-Apr 17 50 Call 15.30% 3286 10370 @ 0.3

Coca-Cola Company (The) Option Trade with 1X Entry

Debit Profit Max Profit Max Risk Delta (Shares) Gamma Vega Theta $30.00 $-5.00 Unlimited $-30.00 27.9 20.7278 $3.55 $-1.64

50.30 50.30 Unlimited% Unlimited% Downside Breakeven Upside Breakeven Max Profit/Max Risk Max Profit/Debit

KO Example • Although the information is for a call option, let’s find the price

of a put option based on this data. • Using the Black-Scholes model, get the price of the put option

assuming: – S =48.99 – K = 50.00 – r = 4% – σ = 0.153 – Days per year = 250 – Days to expiration = 13

( ) 2

1 2 1

ln 2

S r T t

K d d d T t

T t

σ

σ σ

  + + −       = = − −

−

( ) ( ) ( )( ) 2 1, N Nr T tP S t Ke d S d− −= − − −

KO Example

( )2

1

2

0.15348.99 13 ln 0.04

50 2 250 0.5078

13 0.153

250

13 0.5078 0.153 0.5427

250

d

d

    + +         = = −      

  = − − = − 

 

( ) ( ) 13

0.04 25050 N 0.5427 0.3279 48.99N 0.5078 1.2340P e

 −    = = − =

S (stock price) 48.99 K (strike price) 50.00 r (annual rate) 4.00% σ (annual) 0.153 Days per year 250 Days to expiration 13

T-t 0.0520 d1 -0.5078 d2 -0.5427 N(-d1) cum norm dist 0.6942 N(-d2) cum norm dist 0.7063 P 1.2340

ORB 14 Day Put Option Lattice

Day 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Option Value 1.088 0.829 0.597 0.400 0.243 0.129 0.056 0.018 0.003 0 0 0 0 0 0Up

1.346 1.059 0.793 0.556 0.357 0.202 0.094 0.032 0.006 0 0 0 0 0 1.631 1.323 1.028 0.754 0.511 0.308 0.156 0.059 0.012 0 0 0 0

1.936 1.616 1.300 0.996 0.712 0.459 0.252 0.105 0.024 0 0 0 2.255 1.929 1.602 1.278 0.962 0.665 0.399 0.185 0.048 0 0

2.579 2.253 1.924 1.591 1.258 0.928 0.611 0.321 0.095 0 2.902 2.580 2.253 1.922 1.585 1.244 0.898 0.546 0.19

3.222 2.905 2.583 2.256 1.924 1.588 1.246 0.90 3.537 3.225 2.907 2.585 2.258 1.927 1.59

3.848 3.540 3.227 2.910 2.588 2.26 4.154 3.851 3.542 3.229 2.91

4.456 4.157 3.853 3.54 4.754 4.459 4.16

5.047 4.76 5.34Down

Day 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Option Value 1.092 0.832 0.599 0.401 0.244 0.129 0.056 0.018 0.003 0 0 0 0 0 0Up

1.350 1.062 0.795 0.558 0.358 0.202 0.095 0.032 0.006 0 0 0 0 0 1.636 1.327 1.031 0.756 0.512 0.309 0.156 0.059 0.012 0 0 0 0

1.943 1.621 1.304 0.999 0.714 0.460 0.253 0.105 0.024 0 0 0 2.263 1.935 1.607 1.282 0.965 0.666 0.400 0.185 0.048 0 0

2.589 2.261 1.930 1.596 1.262 0.931 0.612 0.322 0.095 0 2.912 2.589 2.261 1.928 1.590 1.248 0.900 0.547 0.189

3.231 2.912 2.589 2.261 1.928 1.590 1.248 0.900 3.545 3.231 2.912 2.589 2.261 1.928 1.590

Days per year 250 3.854 3.545 3.231 2.912 2.589 2.261 4.159 3.854 3.545 3.231 2.912

4.460 4.159 3.854 3.545 4.756 4.460 4.159

5.048 4.756 5.336Down

European Put Option

American Put Option

ORB 14 Day Black-Scholes

( )2

1

2

0.229724.10 14 ln 0.0122

25 2 250 0.6346

14 0.2297

250

14 0.6346 0.2297 0.6890

250

d

d

    + +         = = −      

  = − − = − 

 

( ) ( ) 14

0.0122 25050 N 0.6890 24.10N 0.6346 1.0861P e

 −    = + =

S (stock price) 24.10 K (strike price) 25.00 r (annual rate) 1.22% σ (annual) 0.2297 Days per year 250 Days to expiration 14

T-t 0.0560 d1 -0.6346 d2 -0.6890 N(-d1) cum norm dist 0.7372 N(-d2) cum norm dist 0.7546 P 1.0861

How do we compare with the lattice?

• The American Put option price from the 14 day lattice is:

1.092

• The European Put option price from the 14 day lattice is:

1.088

• The option price from Black-Scholes is:

1.086

Let’s go to our Excel file, then talk about the Greeks!

The Greeks - Delta • It measures the sensitivity of the price of an option with respect

to changes in the underlying security. – In other words, is the rate of change of the option price.

( ) S

tSf ∂

∂ =∆

,

S f

∆ ∆

=∆

( )1N d∆ =

• It is often approximated by

which represents the slope or rate of change between the price of the option and the underlying stock.

• For example, with respect to call options, a delta of 0.7 means that for every dollar the underlying stock increases the call option price will increase by $0.70.

• For a call option the delta can be taken right from the Black Scholes formula:

( ) ( ) ( ) ( )1 2 ,

N Nr T t f S t f

S d Ke d S S

− −∂ ∂  = − ∂ ∂

Delta • Delta can be used to construct portfolios that hedge against

risk. • For example, a dealer (seller) of an option can protect herself

against sudden changes in the price of the stock by buying delta multiplied by the quantity of options sold.

• Rate of change of the option price:

( ),f S t

Hedging with Delta • This indicates maintaining a delta neutral position portfolio.

• A share has a delta value of 1 if its value rises by $1 for every $1 rise in the stock.

• If you own 100 shares of a stock, you can obtain a delta neutral position by buying 2 contracts of the put options with delta value of –50 per contract.

100 – (2 x 50) = 0 Delta

Gamma • The amount of rebalancing required is determined

by Gamma which is defined as:

( ) tS

dN σ

1=Γ

( )2 2

,f S t S

∂ Γ =

∂

• A seller of put options would face a negative Gamma, and buyer of puts would acquire a positive Gamma.

• A higher Gamma means a greater potential loss for the seller and greater potential gain for the buyer.

• A call option gamma can be computed from Black-Scholes as:

• Interpretation of Gamma for a delta neutral portfolio:

• Gamma is the rate of change of delta with respect to the price of the underlying asset.

Theta • Theta measures the change in value of the

option with respect to time. ( )

t tSf

∂ ∂

=Θ ,

• Theta can also be understood as the time decay on the value of an option.

• Assuming everything remains constant, then the option will lose value as time moves closer to expiration.

• For example, if the strike price of an option is $550 and Theta is $20, then in theory the value of the option will drop $20 per day.

• Option price as time passes:

Vega • Is the rate of change of the value of a derivatives

portfolio with respect to volatility: C

ν σ ∂

= ∂

• Vega for all options is always a positive number because options increase in value when volatility increases and decrease in value when volatility declines.

• Decreases towards 0 as the option moves deeper in the money or farther out of the money.

• Vega shows the theoretical price change for every 1% point change in volatility.

• For example, if the price is $7.2 and the Vega is showing 0.35, then as the volatility moves from 10% to 11%, the theoretical price will increase to $7.55.

• Vega vs Underlying Price:

Rho • Rho is the rate of change of the value of a derivative with

respect to the interest rate.

• The value is denoted as the change in price of the option for a 1% movement in the interest rate.

• For example: – A call option with a value of $8.30 is showing a Rho value of

0.45. – If interest rates increase from 2% to 3%, then the price of the

call option will increase to $8.75.

• Rho is larger for options that are in the money and decreases as the option moves out of the money.

• It also decreases as the option gets closer to expiration.

The Greeks Calls Puts

Delta

Gamma

Vega

Theta

Rho

( )1dΦ ( )1 1dΦ −

( )1d S T t φ σ −

( )1S d T tφ −

( ) ( ) ( )1 2 2

r T tS d rKe d T t

φ σ − −− − Φ −

( ) ( ) ( )1 2 2

r T tS d rKe d T t

φ σ − −− + Φ − −

( ) ( ) ( )2 r T tK T t e d− −− Φ ( ) ( ) ( )2

r T tK T t e d− −− − Φ −

( ) ( ) ( ) ( )

1 1

1 1

N standard normal cumulative density fuction

standard normal probability density fuction

d d

d N dφ

Φ = =

= =

Example

Orbital Sciences Corporation Option Trade with 1X

Position Num OptSym Expire Days Strike Type IV Vol OI Entry

Buy 1 ORBDE 8-Apr 18 25 Call 43.90% 18 565 @ 0.75

Debit Profit Max Profit Max Risk Delta

(Shares) Gamma Vega Theta

$75.00 $-30.00 Unlimited $-75.00 37.7 15.6475 $2.03 ($2.56)

• The closing price for Orbital Sciences (ORB) on march 31, 2008 was of $24.10.

• Consider the following call option: – Strike price = $25.00 – Expiration date April (third Friday = April 18) – Days remaining = 14 (trading) days ∼ 3 weeks. – Information provided in Yahoo:

Downside Breakeven Upside Breakeven Max Profit / Max Risk Max Profit / Debit

25.75 25.75 Unlimited% Unlimited%

Example S (stock price) 24.10 K (strike price) 25.00 r (annual rate) 1.22% s (annual) 0.4390 Days per year 250 Days to expiration 14

Call Put Delta 0.384 -0.616 Gamma 0.153 0.153 Theta -8.645 -8.340 Vega 2.179 2.179 Rho 0.483 -0.916

Debit Profit Max Profit Max Risk Delta Gamma Vega Theta $75.00 $-30.00 Unlimited $-75.00 37.7 15.6475 $2.03 ($2.56)

T-t 0.0560 d1 -0.2944 d2 -0.3983 N(d1) 0.3842 N(d2) 0.3452 C 0.6355

N(d1) pdf 0.3820

Portfolio Qty Delta Portfolio Delta Call Options 100 0.384 38.422 Put Options 62.40 -0.616 -38.422 Total 0

Back to our Excel file!

Option Calculators

There are many out there!

Go find a few and compare your results with theirs!

Assignments • Set up your own Black-Scholes put option pricing spreadsheet.

• Use the Black-Scholes equation to solve for the Dell option price; data can be found in

– “Lecture 21 Black-Scholes Put Option Pricing Examples.xlsx”

– How did this compare to the lattice solution that you solved earlier?

– Calculate The Greeks based on the Dell option data.